On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}_p$-extensions of algebraic number fields

All, Timothy; Waller, Bradley

doi:10.5802/jtnb.968

On a construction of

C^{1} (ℤ_{p})

functionals from

ℤ_{p}

-extensions of algebraic number fields

All, Timothy ¹ ; Waller, Bradley ²

¹ 301 W. Wabash Ave Crawfordsville, IN 47933, USA
² 231 W. 18th Ave Columbus OH 43210, USA

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 29-50.

Résumé
Abstract

Soit $k$ un corps de nombres et $k_{\infty} / k$ une $ℤ_{p}$ -extension. Nous construisons un $ℤ_{p} 〚 T - 1 〛$ -morphisme naturel de $lim_{\leftarrow} k_{n}^{\times} \otimes_{ℤ} ℤ_{p}$ dans un sous-ensemble particulier de $C^{1} {(ℤ_{p})}^{*}$ , le dual de l’espace vectoriel sur $ℂ_{p}$ des fonctions continûment dérivables de $ℤ_{p} \to ℂ_{p}$ . Nous appliquons les résultats au problème d’interpolation des sommes de Gauss attachées aux caractères de Dirichlet.

Let $k$ be any number field, and let $k_{\infty} / k$ be any $ℤ_{p}$ -extension. We construct a natural $ℤ_{p} 〚 T - 1 〛$ -morphism from $lim_{\leftarrow} k_{n}^{\times} \otimes_{ℤ} ℤ_{p}$ into a special subset of $C^{1} {(ℤ_{p})}^{*}$ , the dual of the $ℂ_{p}$ -vector space of continuously differentiable functions from $ℤ_{p} \to ℂ_{p}$ . We apply the results to the problem of interpolating Gauss sums attached to Dirichlet characters.

Reçu le : 2014-11-24
Révisé le : 2015-07-30
Accepté le : 2015-07-31
Publié le : 2017-01-27

DOI : 10.5802/jtnb.968

Classification : 11R23
Mots clés : distributions, $L$-functions, Gauss sums, class group

Affiliations des auteurs :

All, Timothy ¹ ; Waller, Bradley ²

¹ 301 W. Wabash Ave Crawfordsville, IN 47933, USA
² 231 W. 18th Ave Columbus OH 43210, USA

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     title = {On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}_p$-extensions of algebraic number fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {29--50},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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All, Timothy; Waller, Bradley. On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}_p$-extensions of algebraic number fields. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 29-50. doi : 10.5802/jtnb.968. http://www.numdam.org/articles/10.5802/jtnb.968/

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